To summarize, the exact \(g\) -function, or the boundary entropy, is defined as the overlap between the finite volume vacuum state and the boundary state. To extract important information about the boundary, we have subtracted a universal constant part \(f_a R\)  from the naive overlap \(G_j^{0}\) , in the definition of \(g\) . The exact \(g\) -function is an important quantity of interest in QFTs with boundaries. It describes the boundary degrees of freedom [1]} and has interesting properties along RG flow [2]}.
